Let be a cyclic group (finite). And let . It means the has a generator that is for some .
Let thus, for .
We need to find the smallest such as,
By the properties of cyclic groups and the fact that is a generator it is equivalent to saying divides .
Thus, we need the smallest such that,
is an integer.
We can write it as, (by dividing by
But,
Thus,
divides .
The smallest such is of course,
Thus, the order of any element is,